practical yield curve models
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practical yield curve modelsTRANSCRIPT
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Developing a Practical Yield Curve Model: An Odyssey
M.A.H.Dempster*, Jack Evans+ & Elena Medova*
* University of Cambridge & Cambridge Systems Associates Limited, Cambridge, UK
+evalueFE Limited, London, UK
Contents
1 Introduction and Background
2 Literature Review
3 Model Evaluation
4 Our Solution
5 Conclusion
References
First Draft 20th July 2012
This Draft 30th September 2012
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1 Introduction and Background
What happens if one wishes to implement an interest rate model based on current knowledge about the subject? Here current knowledge means state of the art or best practice, but not cutting edge. What would someone familiar with the recent literature and current practice be led to? The purpose of this article is to identify and describe the challenges presented by such an exercise.
The background of the exercise is the creation of a new global capital markets econometric model taking account of current developments, but in the spirit of earlier models developed for various financial services institutions, see, for example, Mulvey and Thorlacious (1998) (Towers Perrin), Dempster and Thorlacious (1998) (Swiss Re), Dempster and Arbeleche Grela (2003) (UniCredit) and the current models in use by leading actuarial consultants. The emphasis in these models is the ability to simulate forward over long horizons for pricing various financial products (Dempster et al., 2010), providing investment advice (Medova et al., 2008) and asset‐liability management (Dempster et al., 2003). Since the pioneering system described in Mulvey and Thorlacious (1998), the key to accurate scenario generation from these models are the yield curve models which forecast interest rates in each currency and upon which the determination of all other variables depend.
An immediate first question is what is meant by an interest rate model. Clearly the term covers many different creatures and so we will first describe some requirements for the exercise. However we also hope to illustrate the challenges involved in moving from naive desiderata, through practical specifications, to evaluation of some models described in the current literature. There are challenges at each stage.
Identifying what output is required from the model is probably the most straightforward aspect. After that there is a compromise between hope and reality to be made, but only with a lot of work the two communicate. Some of the issues are well known. For example, two major groups of models may be distinguished by whether they concentrate on matching a wide range of market prices for interest rate related products or instead aim to reflect realistic properties of the time series of rates and phenomena such as risk premia. This is not a logical distinction, but it is already a practical one in that a model builder cannot know what they need to address without considering the actually available models. Thus specification, even at this high level, is already dependent on what models are actually available and the model builder, like the builders of the channel tunnel, must start from both ends and hope to meet in the middle. This is a common enough situation in life, but unfortunately in the present case there is no “Which Dynamic Term Structure Model” publication with a handy feature comparison table to save the model builder from having to read the literature on every model (although James and Webber [2000] is very helpful on the earlier literature).
The lack of such a current guide is not unexpected, but there is no good reason why, to the best of our knowledge, such a survey article does not exist. Unfortunately, the situation is more challenging than simply having to read about each model individually, although there are certainly important features that can be inferred from reading the literature. For example, some models are intended for derivative pricing (e.g. de Jong et al. [2001]) while others are intended for econometric study (e.g. Nelson and Siegel [1987] and their descendents), although upon close examination this
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distinction becomes somewhat murky. Nevertheless, some important features we require are not addressed in the literature at all. The models therefore cannot be assessed on our requirements without first devoting significant effort to implementation.
The object of the exercise that led to this article was to understand how current best practice in the implementation of a dynamic term structure model of interest rates would look. A subsidiary question was whether or not an older model should be updated in the light of new developments. The model arrived at in this paper is be used to simulate long term fixed income returns and to provide realistic scenarios for the future prices of interest rate linked liabilities. In the context of the model, risk premia should be realistically reflected and rates and returns in all scenarios or paths considered should be plausible when taken at face value. In addition, if at all possible the parameters of the model should be robustly determined by consideration of historical data. This is by no means a complete specification of such a model but, since we have already mentioned that there appears to be a distinction between econometric and derivative pricing models, we must accept that we are unlikely to be able to get every feature that we might want in a single model.
The remainder of this paper is structured as follows. The next section contains a brief review of the current literature, followed by Section 3 which describes the process by which we arrived at our final model choice, described in Section 4. It is these two sections which we hope will provide the interested reader with some suggestions and caveats. Section 5 concludes, hinting at the issues involved in embedding yield curve models in the full capital market model for a currency area.
2 Literature Review
As one might expect for such an important and complex subject, the literature on interest rate modelling is vast, and we cannot pretend to have read and processed it all. This article is about the search for an interest rate model suitable for our purposes and, as mentioned previously, we found the widest range and most open ended discussion in the book by James and Webber [2000]. This book also has useful discussions on a wide range of implementation issues. The vast number of options available makes model selection a key issue; there is accordingly much discussion of it in the literature. Unfortunately, from our point of view most of this discussion is focussed on areas that are not directly relevant to our situation. One paper that does address some of our concerns is Nawalkha and Rebonato [2011] which makes clear some of the limitations of derivative pricing models when they are applied for other purposes. These issues led us to consider instead models oriented towards econometric estimation, and affine models in particular. What is involved in affine models was very clearly laid out by Dai and Singleton [2000], who put many affine models into a useful context. However, although their paper is very suggestive, estimation of these models is a much richer topic than a casual reading might lead one to expect. An overview of subsequent developments is supplied in Piazzesi [2010] and vital help with estimation comes from the trio of papers by Hamilton and Wu [2011], Joslin, Singleton and Zhu [2011] and Bauer, Rudebusch and Wu [2011]. These three papers provide robust and efficient estimation methods which provide a firm foundation for the conclusions derived from our explorations. Ultimately, we reject straightforward affine models for our purposes. Instead we pursue the idea of Fisher Black [1995] which has been
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implemented previously in lower dimensional models by the Bank of Japan (see Ichiue and Ueno [2005] and Ueno et al. [2006] and Kim and Singleton [2011]).
First challenge: Lots of models
More than a decade ago, James and Webber (2000) were already able to describe tens of interest rate models. In the intervening years those that might prima facie be suitable for our purposes have not been whittled down to a more manageable number. The range of options is huge and this means that choices may need to be made before some avenues are fully explored.
The lack of a comprehensive current catalogue of models is not merely an inconvenience, it is also indicative of the narrow application of many yield curve models, i.e. their understanding tends to require a narrow focus on the original application of the model. This means that someone seeking a model for a non‐ standard purpose will have to consider potential models in some detail in order to determine their suitability. It also unfortunately means that there are limits to the extent that the required specification can be set down in advance: there is simply not enough information available about what is possible. Specification is a challenge to begin with, and the lack of comparison information in the current literature makes it difficult to understand what compromises will be required, or indeed available. This means that the task specification must necessarily be incomplete before the actual search begins.
On one hand, the specification must start incomplete and be improved upon as information about the relevant capabilities of models becomes available and their unexpected features revealed. On the other hand, candidate models must be implemented far enough to assess their relevant properties. As investigations from the two sides make progress they can both be extended in light of new information discovered and, with luck, be persuaded to meet in the middle.
What do we require? Here the task is to produce realistic stochastic forecasts of the long term behaviour of various fixed income investments and interest rate linked liablities. That means that, among other things:
• Rates must be realistic in each scenario, not only in aggregate
• Returns must be realistic given the economic environment
• Returns should reflect any market risk premium
• The model should consistently address a wide range of fixed income products,
e.g. bonds of different maturities
• Parameters should be objectively and empirically determined
• Simulations should accurately reflect initial conditions.
These requirements are pretty vague, but making them less so is as much a part of the modelling process as achieving the result of our investigations. They will be made more precise and concrete when confronted with specific models. This list is also likely far from complete, even if any particular issue could be put under one or more of the above umbrellas.
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To give an example of what is meant by “realistic” above, the popular Libor Market Model (see e.g. de Jong et al. [2001]) can, under current actually occurring low interest rate conditions, require parameter estimates that make volatility so high that simulated cash returns are more volatile than actual equity returns with significant weight given to rates of more than 10,000%. Nevertheless, the model still works for its intended purpose because in a derivative pricing situation in banks it only needs to have the correct aggregate behavior – which in our case is not adequate.
What’s on offer? It helps to attempt to categorize models first, however crudely. We have seen that for interest rate models the two most important areas are econometrics and derivative pricing. A case can be made that there is a third area, the descriptive models, such as the original Nelson Siegel model. These provide functional forms of yield curves that give economic descriptions of yield curves without direct consideration of the dynamic relationships they clearly imply. Similar requirements are heavily used in both of the other camps ‐ for derivative pricing to efficiently interpolate yield curves from real price data, and in econometrics to provide convenient factor breakdowns of interest rates. However, as with the two main camps, our modelling case needs an explicit dynamic representation, so that a descriptive model can only be a part of our solution at best.
Derivatives pricing is a vigorous field which is intrinsically practical in its focus. That should be good news for us, but unfortunately the imperatives of derivative pricing are somewhat different to ours. In many applications a derivative pricing model must exactly match a wide range of market prices. To achieve this the models frequently incorporate two features that make them very difficult to use for our applications. The first is that they are mostly based on risk neutral pricing and pay no attention to the nature of any risk premia involved. Incorporating a risk premium therefore means extending the model in a typically uncharted way. To apply to our situation many of the implementations will not work, or have to be completely reworked. A more subtle issue is that models are frequently given multi‐parameter term or cross sectional structures which are determined outside their dynamics in order to achieve accurate pricing. This makes such models hard to reason about and estimate statistically. This issue is discussed extensively by Nawalkha and Rebonato [2011] from a slightly less specific point of view.
For the issue at hand the requirement for our model to be objectively and empirically determined translates into the following two more specific requirements:
• Parsimony
• Time homogeneity.
As always we have limited data, so we first want the models to be parameterized by an appropriately limited number of parameters. Secondly, time homogeneity can be seen as a particular aspect of parsimony, but it also means that a model with this property makes it easier to understand what is going on inside the model, since it is more self contained than the double plus models described by Nawalkha and Rebonato. The specification of their models removes a significant part of each model’s ability to reflect market conditions. In practice, this is accommodated by appropriately adjusting a string of model parameters.
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Affine models Econometric interest rate modelling is dominated by affine models. This popularity reflects the attractive combination of tractability and a rich range of specifications. In certain forms they are closely linked to other econometric work horses such as vector auto‐regression (VAR) and an impressive amount of work has been done to apply these models to a wide range of situations. They owe their tractability to how simply, quickly and accurately expectations, and therefore implied bond prices and yields, can be calculated, making it easily feasible to build models of changing expectations. Such models specify transition probabilities between states in a (usually) finite‐dimensional state space and allow expected, or mean, future states to be calculated explicitly analytically or in terms of easily solved ordinary differential or difference equations.
The affine property has been employed in many models, including the famous seminal works of Vasicek [1977] and Cox, Ingersoll and Ross [1985]. However, the links between the various specifications were not systematized until the work of Duffie and Kan [1996] and Dai and Singleton [2000], and their collaborators. From the point of view of someone trying to narrow their modeling options, it is especially helpful that they provide a classification of affine models into a two parameter family. One parameter represents the dimension of the model state space, while the second parameter is the number of those dimensions that are bounded. Roughly speaking, the evolution of the unbounded dimensions follow some sort of Ornstein Uhlenbeck process, while the bounded dimensions follow a Cox‐ Ingersoll‐ Ross, or Feller square root, process. The latter are a priori appropriate for modeling quantities that must remain positive, for example, volatility or interest rates. Of course the details are more complex, but this is the picture that emerges from the literature.
More concretely, we represent the processes to be modelled in the notation of Dai and Singleton’s paper. There they are presented as solutions Y to stochastic differential equations (SDEs) of the form1:
Θ Σ , 1
where Y is the n‐dimensional state vector, W is an n‐dimensional Brownian motion, Θ is a fixed point in the Y (Euclidean) space, and Σ are matrices and S is a diagonal matrix with each entry a constant affine function of Y. The (instantaneous) short rate r is then given by an affine function r of Y, with longer term rates being calculated as expectations of integrals of r.
In most cases we will work with two Y process measures, one describing the actual market evolution of the process and the other assumed for the purpose of pricing. These two specifications, P and Q, are usually termed real world (market) and risk neutral (pricing) measures respectively, and are determined by instances of (1) which differ in their and Θ terms.
Specifically, zero coupon bond prices, or discount factors, are given in terms of Y under the pricing measure Q as
1 We indicate random entities, here conditionally random, using boldface.
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0 '
( ) [ e x p ( ) ] ,
Q Qt Y t
tQt t st
P E d sτ
φ φ
τ+
= +
= − ∫
r Y
r (2)
where prime denotes transpose, t is current time, τ denotes maturity and QtE denotes conditional
expectation under Q at time t. These expressions indicate the origin of the term (exponential) affine model.
Equivalent yields to maturity for all maturities τ, which we will term rates in this article, are given from bond prices by
( ) : lo g ( ) / .t ty Pτ τ τ= − (3)
The key feature of affine models is that for appropriate Y process descriptions the expectations in the bond prices of (2) are easily calculated, either in closed form or numerically. As noted above, bond prices or their corresponding rates may be evaluated for some models analytically, see e.g. James and Webber [2000]. For others, the n vector of rates R may be calculated in terms of the n state variables Y by solving numerically an n x n matrix Ricatti equation of the form
1( ) / ' ( ) ( ) ' ( ) ' 1,2t t t t tR K R R S R rτ τ τ τ τ∂ ∂ = − Σ Σ + (4)
with initial condition (0) 0,tR = where 1 denotes the n unit vector. Expression (4) is an ordinary
differential equation (ODE) with respect to rate maturity τ with other parameters determined by the specifications of (1) and (2).
The role of the affine function in determining the short rate highlights the fact that there will typically be multiple models that give indistinguishable rate evolution processes; in econometric terms these models are unidentified. One of the main contributions of the Dai and Singleton paper is to specify a normalized form for affine models, at least for those models with maximum freedom.
There are similar formulae for state processes involving jumps, which is a subject for our future investigation. More useful from an estimation viewpoint is the existence of discrete time analogues for even complex versions of the state process evolution of these models. These analogues can sometimes considerably simplify the discretization always necessary for estimation. In these cases, the ordinary differential equation (4) for numerically calculating expectations can be replaced with a difference equation which is much easier to solve. This approach is described in Dai, Le and Singleton [2010].
There are many more affine model specifications than those by Dai and Singleton in their categories, but they should all be nested within at least one such category, so these authors’ classification is a good place to start a model search. Ideally, we would examine all affine models systematically, but that is not possible – even in principle. Indeed, the practicalities are not so well worked out that there is an effective algorithm for selecting the best specification. Of course various statistical information criteria and other model selection tools can be applied, but doing so remains something of an art form.
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Instead we start by taking a lead from the work of Litterman and Scheinkman [1991]. They performed a principal components analysis of US yield curve (rates) data and concluded that at least three factors were necessary for a satisfactory representation of the behaviour of yield curves. Since we are also determined to avoid complexity, we should also avoid using more than three factors if at all possible; so these models are our starting point.
3 Model Evaluation
Basic considerations
The analysis presented here is based on data on gilt yields from the Bank of England web site. The data series extend from 1972 to the present and are freely downloadable. Among the series available are zero coupon yields (rates) for maturities at one month intervals at the short end and at six monthly intervals out to up to 30 year maturity. This data is an excellent resource for development, but it is not ideal and may not be what we use in production. Its shortcomings include the fact that interpolation and coupon stripping is done using a model which is not necessarily consistent with the ones that we are evaluating, and which cannot distinguish “on the run” effects and other similar microstructure considerations. There are also gaps in the data, although to be fair these mostly correspond to periods when certain maturities did not exist or were not liquid. Nevertheless, full zero coupon yield curve data at daily frequency extending as far back as the demise of Bretton Woods is extremely useful and is sufficient to address all the issues tackled here. We shall in fact use end of month data at monthly frequency for the investigations reported in the sequel.
Model parameter estimation will be based on some version of maximum likelihood estimation (MLE) or its approximants. To this end we use numerical optimization routines that combine an initial search method with an appropriate quasi‐Newton method to refine the maximum achieved by search. Since in practical use parameter estimates need to be frequently updated, we have a predilection for using market data at only a few data points on the yield curve and then interpolating or extrapolating other required maturities as necessary, e.g. for pricing. If the same number of observed points as the state space dimension is used, as is assumed in (1), (2) and (3) above (here3), we can often considerably reduce the estimation computations by the direct inversion of square matrices.
This rules out, at least initially, most filter‐ or method‐of‐moments‐based estimation of unobservable (latent) factors as too computationally expensive for our application. Moreover, although typically incorporating more observed rate data, all such complex methods require “tender loving care” in each particular data instance, which is a difficult requirement in a production environment.
First candidate specification The Dai and Singleton classification describes four types of three factor exponential affine models classified roughly according to how many of the factors are bounded (as noted above). Which to
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investigate? It would be helpful to have a model that avoids negative rates. That can only happen completely in a model in which all factors are bounded, which means a model of 3 type.
Such models can be thought of as a three factor extension of the well known Cox, Ingersoll, Ross [1985] square root (CIR) one factor model. They have been famously addressed in Chen & Scott [1993], so this should not be entirely unknown territory, and their use guarantees non‐negative rates.
The Dai and Singleton normal form for the stochastic differential equation (1) representation of this type is relatively simple. The matrix must be positive on the diagonal and negative off the diagonal and the parameter vector Θ must have positive elements, but otherwise they are allowed full freedom, while Σ is the identity matrix and S is a diagonal matrix with Y on the diagonal. As a result the instantaneous rate can be given by a linear function of Y and a constant, i.e. an affine function of Y. We calculate model implied bond prices, and the corresponding rates using (3), in terms of the model parameters by numerically solving the appropriate version of (4) to determine the corresponding likelihood function for maximum likelihood estimation with the observed rate data (see e.g. James and Webber [2000]).
For this model we have 28 coefficients to estimate – 12 for the market measure process, 12 for the pricing measure process and 4 for the form of the instantaneous rate. The risk premium is then the difference between the drift terms in the two models. Note however that although there are 28 degrees of freedom, 24 of them are bounded for technical reasons and the other 4 parameters, which correspond to the affine form of the instantaneous rate, must be restricted to be positive for economic reasons.
Unfortunately using an 3 model in practice is not as smooth as we had hoped and problems arose even before estimating the likelihood. For a given parameter set such a model maps the positive orthant of a linear state space to the space of rates through a nonnegative affine map This bounds each rate from below and means that simple stylized characteristics of yield curves, such as its slope, defined as the difference between short and long term rates, are also bounded. This limits the range of possible results, but unfortunately it is not the only restriction. There is a condition on Feller square root processes which must be satisfied if the process is to remain positive and there is also a link between the different rates through their derivation as expectations of integrals of the short rate under the Q pricing measure. Both these restrictions give the resulting model further stiffness.
The consequence is that the range of combinations of rates that can be achieved for a given parameter set is limited, and frequently will not span the set of observations. In particular, very low rates, negative yield curves and observations for which the middle rate was far from a linear interpolation between the long and short rates observed, all provided examples that were outside the span of apparently reasonable parameter sets. This problem is fairly intractable because the parameter sets are also bound by the need to match moments like the standard deviation of the short rate (which provides a lower bound for the mean of the process) and, through these requirements, tight restrictions on the range of parameters and spanned yield curves.
This is a problem on two fronts. On the one hand, the range of yield curves rendered by such a model as not merely unlikely, but actually impossible, is uncomfortably large. On the other hand,
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many parameter sets could not provide likelihoods for all observations. This means that estimation first requires a search for valid specifications and it is not clear that there are any for the data set used in this paper. Unless the other modeling possibilities are even worse, these results imply a pretty direct failure of this model to provide realistic simulated yield curve scenarios.
The problems are however not only on the simulation side. If the model cannot fit even a minimal set of observations exactly, any estimation technique that uses observations directly or infers a state from direct inversion will fail too. Chen and Scott (2003) encountered this problem, as their comparisons between observed and fitted yield curves show. To get around the difficulty this poses for estimation, Chen and Scott resorted to a form of Kalman filter to estimate the expected state evolution path as an input into a maximum likelihood estimation for their state evolution parameters. This adds a whole extra layer of modeling, which from our viewpoint further
undermines the 3(3)A model type as a starting point for this investigation.
Based on the investigation of one model alone – however disappointing the results – there is of course still the possibility that it is the best available. However, with the limitations in the performance of such a model in evidence right from the start – and the requirement for significant extra technical elaboration required even to begin to make progress – it seems reasonable to look elsewhere.
Another candidate Having turned back from the square root process models, an obvious place to continue would be with one of the models examined in the Dai and Singleton [2000] paper. The concrete investigation
in that paper concentrates on models of types 1(3)A and 2 (3)A . Since in general it appears that
Gaussian models are easier to handle than CIR type models, an 1(3)A type model might be
preferred. Dai and Singleton present a restricted 1(3)A model with a pleasing intuitive
interpretation, originally given by Balduzzi, Das, Foresi and Sunderam [ 1996] and henceforth termed the BDFS model. This model proposes a short rate r with a stochastic mean θ and stochastic volatility v . Roughly speaking the log short rate and the mean log short rate follow mean reverting Gaussian processes, while the volatility follows a CIR‐type process, but the short rate is allowed sensitivity to innovations in the volatility.
More precisely, the (slightly extended) model process is given in the market measure P by
0 0
0 00 0
Θ 5
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Σ1 0 0
1 01
0 00 00 0
.
The risk neutral version under the Q measure will have the same form, but with different parameter values for and Θ and so both models have a relatively parsimonious 12 parameters. A model
equivalent to the original BDFS model is given by setting the parameters 1θ , 2θ , and to
zero, leaving only the innovations of r and correlated. For this affine model the short rate and rates are once more available in analytical closed form under the pricing measure Q.
The model again has three factors, so we might once more hope be able to estimate it successfully using either three points (rates) on the yield curve or three principal components from the data. A feature of affine models is that any point on the yield curve is modelled by an affine function of the state space and therefore so is any linear combination of them. Hence the linear combinations of rates that are principal components of a finite data set are also modelled. As a result, a range of choices of estimation strategy are available – we can use a variety of combinations of three points
on the yield curve, or even levels of principal components, as observables.
At first this approach appeared to go quite well. We were able to fit the observations for a variety of specifications, and distributions of the modelled yield curve points (rates) looked reasonable. For
example, in the chart presented below the 1(3)A model was fitted to the 1month, 5 year and 15 year
rates. In general, the model had no trouble fitting the observations and the estimated moments appeared plausible.
So this approach appeared to be quite promising, until we plotted the out‐of‐sample forward rate evolutions implied by the model for various points on the yield curve together with those of the historical data (see Figure 1 grey scale).
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Figure 1 BDFS model implied yield curves
The red lines in Figure 1 are the in‐sample paths of rates linearly interpolated between fitted yield curve points in the data over 18 years from 31st January 1994 to 31st December 2011 , while the blue ones represent the corresponding paths of yield curve points implied from the model, both in‐sample and out‐of‐sample by average values of simulated scenarios. The units of the vertical scale are in discount yield per month, so that the in‐sample implied yield curves have rates from about ‐16% per annum (p.a.) at 18 months, can exceed 13 % p.a. at 10years and can each reach over ‐70% p.a. at 20 years. So not only would the implied yield curve not match observations, but its implications could be wildly implausible.
This chart was an unpleasant surprise, but it demonstrated some of the less obvious challenges involved in implementing an affine term structure model, including but not limited to:
• negative rates
• explosive yield curves
• importance of rates beyond those directly fit to data – both before and after the sample dates.
After seeing this we were struck by how rarely in the literature implied yield curve plots are given over the in‐sample data period, with the honourable exception of Chen and Scott’s 1993 paper.
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The implied yield curves shown above in Figure 1 clearly illustrated subtleties in the modelling that we had missed in our direct approach to estimating the model. Further investigation also revealed rather extreme values of the parameters. In particular, the yield curve blows up both before the sample begins and shortly after the longest observed term.
If the original understanding of this model was that the square root state variable process was meant to capture the volatility of volatility, then the actual estimates are a significant disappointment. These estimation results could be interpreted as confirming the conclusion of Litterman and Scheinkman that three factors are necessary to capture the dynamic behavior of the yield curve, but that stochastic volatility does not appear to be the natural third risk factor. This in turn suggests that a mis‐specification of stochastic volatility is ill suited to match the behaviour of a third yield curve point or principal component.
That being the case, we decided to start at the beginning and work with as simple a model as possible. Within the affine framework that means purely Gaussian or extended Vasicek style models.
Gaussian models
The basic extended Vasicek model, in the Dai‐Singleton terminology type 0(3)A , is given in the
market measure P by (1) with
0 0
0
Θ
Σ0 0
0 00 0
6
1 0 00 1 00 0 1
.
Estimating a complete specification means identifying 16 parameters.
This specification is not intended to restrict the process, but rather to represent a family that spans the range of specifications which are not equivalent. Since we wish to allow the broadest range of risk premia, to fix the process in the pricing measure Q in the form above we must allow non‐zero lower triangular values of but zero the values of Θ. We assume a fixed risk premium and so that market and pricing measure process SDEs differ only by the value of Θ, giving the risk neutral model 13 parameters.
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Such a model and its variations are much easier to work with than other affine models.
They have closed form formulae for the distribution of the state over any time period and there is a big overlap with standard econometric techniques. It should therefore be easier to isolate and address the issues previously identified with the BDFS model.
With hindsight, this would have been the right place to start, but it was hard to identify the challenges likely to be posed by attempts to implement superficially more attractive models. There are of course still many choices to be made with this model, and other issues to address, but the behaviour of 0(3)A models should be much more predictable.
Identification One major change in going from the BDFS model to a pure Gaussian framework is that the default model is now unrestricted. That does not cause any immediate operational difficulties, but the initial results were not promising. Actually running the model does not cause problems and the behaviour of the directly calculated quantities looks reasonable, but plotting the implied yield curve paths (see Figure 2 grey scale) once again reveals problems.
Figure 2 Vasicek model implied yield curves
(8)
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The yield curve makes even more violent swings than for the BDFS model, suggesting two year rates of over ‐20% p.a. and 10 year rates of over 60% per annum. At the time of writing there are some people talking as if these were in fact possible alternative scenarios, but this seems unlikely to us.
So, what is the problem? One observation is that in the analysis applied we paid no attention at all to the behaviour of the rates between the modelled yield curve points, so we have only these three points to rely upon to deliver reasonable interpolations.
A more subtle issue lies within the estimation algorithm itself. The basic idea is that, given a parameter set, we can identify the state from each observation and determine a likelihood of each transition of the corresponding process which we can then use as a basis for likelihood optimization. The problem is that we need to do this in terms of the market (physical) measure, while we can only determine the expectations needed to compute rates or discount factors in terms of a pricing (risk neutral) measure. Working with the above assumption that the difference between the two is a constant vector in the state process drift, as implied by (6), we can obtain its estimate.
However, the observations are not used to make a direct forecast of short term rate changes and the model therefore has the freedom to trade off between the risk premium and the forecast rate change. This freedom leaves us with an unidentified model; but a lack of identification that is not transparently obvious. Until recently this point was little discussed explicitly in the literature, presumably because much of the parameter estimation for these models often works with an over‐identified Generalised Method of Moments (GMM) approach, which eradicates the problem.
In this instance unrealistic dynamics highlighted the inadequacy of our estimation procedure, and thus the lack of identification. Although the surprisingly slow convergence of the optimization calculations involved might also have provided a clue, this shortcoming was not immediately evident from the results.
Care in likelihood optimization The issue of identification and estimation is addressed in some detail in the recent papers of Hamilton and Wu [2011] (HW) and Joslin, Singleton and Zhu [ 2011] (JSZ). The estimation procedure is completed by requiring that in addition to the observations fitted exactly by the affine model; additional points are fitted approximately using a generalized least squares criterion. This extra information is enough to tie down the interaction between the dynamics of the model and the risk premium, even when the specification of risk premia is expanded to allow the difference between the market and pricing measures to be an affine function of the state, rather than merely a constant. This broader specification of the risk premium allows statistical consideration of the fit of the model for points on the yield curve between perfectly fitted rates and also goes some way towards addressing the issue of return prediction raised in Cochrane and Piazzesi [2005].
Although the stochastic differential equation representation (1) of this model is familiar and suggestive, it does obscure a couple of its subtle points. Representations of multi dimensional Gaussian processes in this framework demand the square root of a matrix which introduces extra degrees of freedom having no obvious meaning in the model. These can to some extent be dealt with by normalization, but the normalization may obscure the symmetries of the model. A second
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point is that the model is fundamentally a state space model with observable rates (yields) y in a linear state space.
Although we do not propose to introduce the full relevant notation here, it can be useful to think in terms of the physicists’ bra and ket terminology. In these terms a state is a bra vector and an observable is a ket function, which gives a specific value when evaluated at a particular bra vector. A set of independent ket vectors, or observables, can thus be thought of as coordinates on the bra space. We can therefore parameterize the state space in terms of given observables. If these are affine functions of the state vector , the model can be expressed in terms of the observables and it is again an affine model with parameters that differ from those of the original expression by application of a known affine transform.
In affine models expectations of affine observables are again affine and as a result we can without loss of generality express an affine term structure model in terms of rates, or even the kind of linear combination of rates that results from a principal components analysis of yield curve observations. This brings term structure models much closer to traditional econometric methods and also means that different estimations can be expressed in terms of equivalent parameterizations for comparison purposes.
In our situation this means the state vector can be expressed in terms of specific observables and the discretization of the 3 HW/JSZ model can be expressed in the m vector form
Δ Σ ,
where m n of (1).
The parameter vector contains only the long run means of the rates (or principal components) chosen to be fit to the data exactly, their mean reversion rates and the rate covariance matrix multiplying the independent uncorrelated standard normal Gaussian innovations vectorε . The exactly fit rates are specified by appropriately specified zeroes in the full mxm Σ matrix. Here we will set m := n+1=4.
The log likelihood of the transitions of this vector auto‐regression (VAR) model can therefore be expressed as a sum of squares and additional terms that depend upon the state risk premia and the affine expression for the instantaneous short rate. The overall parameter estimation can then be performed in two steps:
• the parameters of the state process under the market measure are first estimated via the simple, quick robust generalized least squares regression procedure,
• then the risk premia are estimated by a slightly more complicated MLE calculation involving one extra observable and a helpful simplification based upon normalization of the drift of the state process under the pricing measure, see e.g. Joslin, Singleton and Zhu [2011] for details.
However, both cited papers also highlight the issue of local optima in this type of augmented likelihood optimization. Indeed, Hamilton and Wu demonstrate that an estimate obtained in Ang and Piazzesi’s highly cited paper (Ang and Piazzesi [2003]) is in fact only a local maximum, and that a global maximum not only has a significantly higher likelihood, but also has opposite signs for key sensitivities.
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Implementing all these considerations using 4 observed rates leads to a considerable improvement for the HW/JSZ model over the previously evaluated models using only 3 rates, but the burden of needing to calculate tens or hundreds of optimizations from different starting points in order to have confidence in attaining the maximum augmented likelihood is unwelcome.
Moreover the results are not automatically a great deal more plausible. For example, the estimation of the model represented above gives the unpleasant behavior shown in Figure 3 (grey scale).
Figure 3 HW/JSZ local optimum model implied yield curves
This chart shows equivalent information to that of Figure 2 for a locally optimal parameter set obtained with the HW/JSZ model incorporating the additions discussed above. The oscillation is caused by complex eigenvalues in the matrix drift component of the model. In a representative maximum likelihood estimation of the parameters of this model (under the pricing measure) which corresponds to a local optimum with maximal real eigenvalue 0.96076, the largest complex eigenvalue is 0.581612 +/‐ 0.815219i.
The steep gradient of the curve near the observation points means that only very small changes in the state are required to deliver the observed movements. Thus the resulting transition densities
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can have values of very large magnitude. There is of course no intrinsic reason that a solution of this type should not be the global augmented maximum, but its implications are of limited practical use to say the least.
High frequency oscillations like those illustrated above can be avoided by some combination of a judicious choice of yield curve points for approximate fitting, bounding the imaginary component of the eigenvalues of the drift matrix, and good fortune, so that the resulting estimates begin to look reasonably behaved. The search for optimal solutions is carried out using several starting points and we have mostly been fortunate in finding that the best of these did not in fact have high frequency oscillations.
However, were we to be confronted with such a result, or if we needed to rely on an estimation technique not producing such an outcome, there are several approaches available to avoid it. The most direct method is to restrict the range of complex eigenvalues allowed. In the JSZ method this is straightforward because it considers the Jordan normal form of the drift matrix and we can apply such limits to the drifts directly. It can also be useful to apply a penalized buffer zone between unrestricted parameters and those absolutely prohibited in order to avoid the likelihood search path falling into a hole.
It is possible to extend this analysis to consideration of more rate data than 4 points on the yield curve, e.g. including more points between the exactly observed points, but as a result the parameter estimation algorithm is made substantially more complex.
Negative yield curves High frequency oscillations can be reasonably straight forwardly suppressed using the techniques suggested above, but it is much harder to avoid estimated negative yield curves in this model. This is not simply the possibility of negative rates, but also the possibility of discount factors greater than 1 over long time periods. In some of the earlier models evaluated this was a problem, but, having by this point eliminated many of the explanations for such behavior, the problem still arises with the HW/JSZ model.
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Figure 4 HW/JSZ long term yield curves
Figure 4 (grey scale)shows more reasonably behaved yield curve evolution (data and model for various yield curve maturities) over the 20 year data period. Up to the observed data horizon (31 December 2011) the model fits the data well, but for the model yield curve forecast at 20 years forward the curve becomes strongly negative. These projected yield curves are derived by simulating (5) beyond the data span, using (6), and then for each maturity averaging closed form model implied rates across a large number (4,000) of simulated scenarios to yield an accurate approximation of rates derived from the continuous time and space zero coupon bond prices (2).
The root of the long term negative yield curve problem is that the weightings of low rate scenarios are much heavier than those of high rate scenarios in the scenario average rate calculations. Although it is not quite inevitable, this problem is obdurate and the restrictions on the combination of mean level, mean reversion and volatility necessary to prevent long term negative rates can be unrealistic for realistic shorter term behaviour.
Explicit or computable formulae for expected rates are one of the key features of affine models, but especially in higher dimensional versions these can be quite opaque and do not immediately yield an intuition about what is going on. The key feature is the distinction between the appropriate mean
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coordinates of the state process: the expected future short rate and the expected forward rate or yield to maturity of the appropriate zero coupon bond. The first is visible in the appropriate Ornstein Uhlenbeck equation and the second takes into account the time asymmetry of discounting, so that low rates count for more than high rates. The expected forward rate, which actually determines the yield curve, also reflects the covariance of the future short rate with the discount factor up to the time point in question, the so called convexity adjustment.
To illustrate this, consider the SDE of a simple one factor model:
,
with : 0.04 , : 0.01 and 0.04 . The short rate increment has a mean of 0.04, a future short rate that is asymptotically zero and forward rates that are negative. Even when the asymptotically future rate is positive, the forward rates can be negative. In fact, doubling the mean reversion rate θ gives negative forward rates despite the positive future short rate, and θ must be increased by a factor of 15 to make the implied yield curve positive in this example.
Our estimates for the JW/JSD model typically suggest very slow mean reversion and therefore a small value of mean reversion coefficients and make an ultimately negative yield curve very likely.
On the other hand, fast mean reversion implies very large risk premia and strong divergence from the rational expectations hypothesis which leads to unit roots under the market measure. Of course the recent regime of historically low interest rates does not help the situation, but the problem appears quite deep seated. Unit root model estimates will cause problems whatever the actual rates are.
Unit root solutions Not only do unit root estimates bring negative yield curves, the range of returns given by a model with a unit root or super unit roots becomes extremely wide. Many examples estimated in the literature have (super) unit roots, see e.g. references. Moreover, we have found that implementations of the small sample correction suggested by Bauer, Rudebusch and Wu [2011] yield adjustments easily large enough to change a mean reverting estimate into one with a random walk (or worse) by introducing a unit root. In a representative estimation of the JW/JSD model this correction changed a maximal real eigenvalue 0.96138 into one of 1.03142. If we follow the authors’ suggestions of applying a shrinkage factor or restricting the search range to mean reverting solutions, these heuristic techniques may result in expunging unit root estimates, but they delegate to other considerations the determination of whether or not the state process has a unit root.
Despite the significance of this issue it does not seem to be directly addressed in much of the literature. Clearly there is typically a problem with the amounts of data available, but the observation that this does not matter much for very short term forecasts does not help determine parameters that have great significance for long term behaviour. The literature frequently defers the decision on the long term stability of the model to outside considerations and in general provides little help in the empirical determination of the estimated state process stability.
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Negative rates As it happens, our best estimates do not suggest unit roots and therefore we are not forced to consider wildly unprecedented risk characteristics. While it is regrettable that we cannot have both our first choice estimates and the ability to extrapolate indefinitely beyond the range of our observations, if we were only considering aggregate statistics over a short horizon we would probably be able to work with the models evaluated so far.
However, we need every simulated scenario to be plausible in its own right, not merely as part of an aggregate. At this point the issue of negative rates becomes a genuine problem.
Figure 5 (grey scale) shows the extent of this problem. It depicts quantiles of simulated ten year interest rates from a recent starting point, 30th December 2011. Nearly 25% of scenarios are negative. If we merely assumed that those scenarios were zero instead of negative, the mean rate would be about 25 basis points higher ‐ an amount too large to ignore. This is unacceptable even when other aspects of the model are well behaved.
Figure 5 HW/JSZ ten year rate distribution out‐of‐sample evolution quantiles
Typical estimation only looks at conditional likelihood Much recent interest rate data appears to have a strong downward trend, reflecting the fact that rates are currently very low. In the United Kingdom they were very high after the end of Bretton Woods and in the late 1980s. If we consider only the likelihood of the transitions, the trend will often
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be made plausible by strong mean reversion effects. Unfortunately, this can result in estimated models that make even recent history practically impossible. This is because analysis of transitions without considering the long run relevance of initial conditions other than the first observation pays no attention to how the world got into the first state.
In terms of maximum likelihood estimation, considering only the transitions is considering only conditional likelihood. General descriptions of MLE and descriptions of univariate MLE usually concern a full or unconditional likelihood function, but for multivariate time series analysis this seems to be rarely considered. For instance, Hamilton’s [1994] book opens discussions of maximum likelihood estimation with a full description of unconditional likelihood in the univariate case, but in the multivariate case it says that unconditional estimation is not attempted.
It is not difficult to derive the expression for the unconditional likelihood when the process is stationary, when the long run mean estimate can be used as the initial state. However using the unconditional likelihood makes estimation more difficult in a number of ways. Moreover, conditional likelihood estimation might be necessary to ensure that the stationarity assumption of rates is reasonable. The first order lags incorporated in the discretized versions of the models considered make their full likelihood expressions much more complicated and standard least squares techniques no longer apply. Nevertheless, in our situation unconditional estimation appears to be the natural approach.
Gaussian models have constant volatility The discussion so far has addressed relatively broad issues with the evaluated models – effectively the minimum requirements needed for viability. At the start of our attempts to implement an interest rate model there were other, subtler, issues on the agenda. One of these is an empirical link between the level of rates and the volatility of rates. The connection is well known and is illustrated in Rebonato’s [2004] book. In stylized terms the behaviour referred to is positive correlation between volatility (measured in absolute terms) and the level of rates. In a purely Gaussian model there is no such link, so volatility will either be too high when rates are low or too low when they are high. This problem is quite subtle – models which have lognormal interest rates have a similar problem of inflexibility, but the sign of this behaviour is reversed. As a result, when historical rates are very low, very high levels of parametric volatility are required to match historical or option implied volatility levels.
Lessons learned Before describing our implemented solution in the next section, we first discuss what we did for each model investigated so far and what we have learned from these exercises.
Draw a chart
Some of the problems we encountered were easily noticed by drawing a graph. Graphs of paths of yield curve points over the data period rarely feature in the literature. The same comment applies to plotting quantiles of the evolving distribution of interest rates and returns. This is easier to do than most of the modelling, and provides extremely useful context for model evaluation.
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Take care of specifications As a rule of thumb, when thinking about how big a model to build, it seems sensible to talk about the dimension of the state space of the model processes, but there is inflexibility in the CIR square root processes that means they do not map well to the three principal components identified in the Litterman and Scheinkman [1991] paper. From this point of view it is probably better to count the number of Gaussian variables in a model.
Use multiple likelihood optimization starting points The log likelihood functions of models of this type have numerous local maxima. A single optimization will most likely not give the global maximum. Unfortunately the parameter search space will typically be quite large and will require computationally intensive multiple optimization starting point calculations, which luckily can be straightforwardly parallelized.
Consider using unconditional likelihood While using unconditional likelihood makes some procedures more difficult or complex from specified initial states, but if the calculations are feasible there seems no reason why this should not be the default procedure. For Gaussian affine processes the computations are relatively simple. Taking unconditional likelihood into account in interest rate series where the rate levels at the beginning and end of the series are very different will tend to yield estimates with greater long term variance. Unfortunate side effects of this are that the sample statistics of the time series are no longer the estimated model parameters and super efficient least squares calculations can no longer be simply applied to calculate the parameter estimates. Even so, the required calculations are not terribly onerous.
4 Our solution
For our application negative yields are a fatal problem. However in Gaussian models they are unavoidable and cannot be sidestepped in forward solutions by simple measures such as setting any negative rate to zero, although some other more complex procedures are available which alleviate, but not eliminate, the difficulty.
Instead we applied an old idea of Fischer Black’s. The idea asserts that interest rates generally come with a “stuff it under the mattress” option in which investors just keep their money when faced with a negative interest rate. This means that we might model interest rates as options on some more fundamental, but possibly hard to observe, shadow rates which guarantee positive rates or set minimum rates.
The original idea is described, but not implemented, in Fischer Black’s last (posthumously) published paper, see Black [1995]. The corresponding yield curve model has been implemented a number of times before, often with Japanese interest rates in mind. Notable examples include papers by a group at the Bank of Japan (see Ueno, Baba & Sakurai [2006] and Ichiue & Ueno [2007]) and by Kim
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and Singleton [2011], but with no more than two state variables. In our case, we used three variables.
Specifically, the underlying process evolves in a 3 dimensional state space like that of the HW/JSZ model, but now the forward yield observables are nonlinear functions of the state space since they are represented in terms of expectations on integrals of an option on short rate affine functions of the state space. As a result, unlike the HW/JSZ exponential affine model, coordinates are no longer uniform over the state space and translation between yields which we can observe and model states about whose probabilities we can reason about is no longer a straight forward calculation.
In other words, we hypothesize an 3 model describing the evolution of a shadow short rate of exactly the form described by (6) in the previous section, but additionally hypothesize that the actual short rate is a zero strike call option on this rate, so that
, max 0, , .
This is conceptually simple, but as noted above the nonlinearity introduces considerable practical difficulty in moving from our yield observations to the corresponding state. Kim and Singleton [2011] perform this calculation by solving a parabolic partial differential equation (PDE) of the form
2 2
2
, 1 1
/ / / 0 ,t ij t i j i t i ti j i
P a P y y b P y cPτ= =
∂ ∂ − ∂ ∂ ∂ − ∂ ∂ − =∑ ∑
with boundary values (0) 0P = and sufficiently large state space bounds for the rates. However the
alternating direction implicit (ADI) finite difference scheme they use to solve (10) does not easily extend to the corresponding PDE in three dimensions and so far we have been unable to implement an effective alternative.
There are still avenues to explore in this direction, but for the time being we have adopted a technique which is a combination of analytical closed form yield calculations and Monte Carlo simulation. The procedure may be considered to be a refinement of solving the least squares fit of Monte Carlo simulated yields to each individual observation. This approach is very computer intensive, but it does appear to provide very robust results.
Outofsample forecasts
We present here the quantiles of the simulated out‐of‐sample evolution over time of the distribution of the 10 year rate from for the new nonlinear Black model. Figure 6 displays the out‐of‐sample diagram analogous to Figure 5 (from the same 30th December 2011 starting point) for this model.
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Figure 6 Black model nonnegative 10 year rate distribution out‐of‐sample evolution quantiles
Figure 7 shows the in‐ and out‐of‐sample long term yield curve simulation forecasts for the new model analogous to that of Figure 4 and demonstrates the required long term scenario simulation performance of the Black model that our applications require.
Besides delivering positive rates and the direct connection between rates and expectations, the most notable difference between our earlier attempts and the Black model seems to be that it allows much slower short rate mean reversion estimates, which is significant. For example, in the model that this research is meant to update (a development of the one described in Mulvey and Thorlacious [1998]) disequilibria have an estimated half life of approximately 4 years, while in this model the estimated half life is nearly 15 years in the physical measure, and over 1,000 years in the pricing measure used for yield curve extrapolation. The difficulties such slow mean reversion would cause in a straightforward Gaussian model have been clearly illustrated above.
In concrete terms, this property of the new model means that we can reflect the possibility that interest rates in our main modelling targets – the UK, US and EU – remain extremely low, as Japan’s have done for some time, while also entertaining the possibility that rates will blow up, like those of Italy and Spain currently. Indeed, the medium term distributions of short rates are bimodal. The underlying (shadow expectation) processes are still Ornstein Uhlenbeck processes, but the way states are nonlinearly mapped to yield curves is different from the affine map of the standard earlier models.
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Figure 7 Black model long term nonnegative yield curves
5 Conclusion
In this article we have described a voyage of discovery that has resulted in a new tractable yield curve model which possesses the properties required by practitioners who intend to use long term simulations from it for pricing, investment advice and asset liability management. Its estimation from sovereign interest rate data is computationally intensive, but straightforward and tractable, and is amenable to frequent updating, as is required by these applications.
Table 1 summarizes the properties of the models evaluated in this paper regarding well accepted stylized facts of yield curves and their models in the order in which both properties and models are discussed in the paper. Our nonlinear Black model possesses all the desired properties except for
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closed form bond prices, but rapid effective PDE‐based numerical techniques for bond pricing have been described in Section 4.
Stylized Fact Properties Yield Curve Model
CIR BDFS Vasicek HW/JSZ HW/JSZ/BRW Black
3(3)A 3(3)A 3(3)A 1(3)A
0(3)A 0(3)A
Mean Reverting Rates Yes Yes Yes Yes No Yes Nonnegative Rates Yes No No No No Yes Stochastic Rate Volatility Yes Yes No No No Yes* Closed Form Bond Prices Yes Yes Yes Yes Yes No
Replicates All Observed Curves No Yes Yes Yes Y es Yes
Good for Long Term Simulations No No No No No Yes
State Dependent Risk Premia No No No Yes Yes Yes +ve Rate/Volatility Correlation No No No No No Yes Effective in Low Rate Regimes No No No No No Yes
Table 1. Properties of evaluated yield curve models with regard to stylized facts
*Rate volatilities are piecewise constant punctuated by random jumps to 0 at rate 0 boundary hitting points.
In ongoing work we are developing the interaction of this model with the other financial and economic variables in interacting capital market models of the four main currency areas. A first step explores the statistical interactions of our new yield curve model with macroeconomic variables such as inflation rate and GDP begun by Ang and Piazzesi [2003] in a macroeconomic context and extended in a manner more appropriate to capital market models by Dempster, Medova and Tang [2012].
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